Succinctness, verifiability and determinism in representations of polynomial-time languages

Several representations of P, the class of deterministic polynomial time acceptable languages, are compared with respect to succinctness. It is shown that requirements such as polynomial running time, verifiability of running time, and verifiability of accepting a set in P can be causes for differences in succinctness that are not recursively bounded. Relating succinctness to nondeterminism, it is shown that P ≠ NP if and only if the relative succinctness of representing languages in P by deterministic and nondeterministic clocked polynomial time machines is not recursively bounded. questions are posed, concerning the implications of P = NP, with respect to translatability and succinctness between other pairs of deterministic and nondeterministic representations for P.